How many sides does a polygon have if the sum of the measures of its internal angles is five times as large as the sum of the measures of its exterior angles ?

Options

A. 20
B. 12
C. 15
D. 10

Answer

To find out how many sides a polygon has when the sum of the measures of its internal angles is five times the sum of the measures of its exterior angles, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formulas**:
- The sum of the internal angles of a polygon with \( n \) sides is given by the formula:
\[ \text{Sum of internal angles} = (n - 2) \times 180^\circ \]
- The sum of the exterior angles of any polygon is always:
\[ \text{Sum of exterior angles} = 360^\circ \]

2. **Set up the equation**:
- According to the problem, the sum of the internal angles is five times the sum of the exterior angles:
\[ (n - 2) \times 180^\circ = 5 \times 360^\circ \]

3. **Calculate the right side**:
- Calculate \( 5 \times 360^\circ \):
\[ 5 \times 360^\circ = 1800^\circ \]

4. **Substitute and simplify**:
- Now, substitute this value back into the equation:
\[ (n - 2) \times 180^\circ = 1800^\circ \]
- Divide both sides by \( 180^\circ \):
\[ n - 2 = \frac{1800^\circ}{180^\circ} = 10 \]

5. **Solve for \( n \)**:
- Add 2 to both sides to find \( n \):
\[ n = 10 + 2 = 12 \]

6. **Conclusion**:
- Therefore, the polygon has **12 sides**.

### Final Answer:
The polygon has **12 sides**.

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